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Theorem mapssfset 8868
Description: The value of the set exponentiation (𝐵m 𝐴) is a subset of the class of functions from 𝐴 to 𝐵. (Contributed by AV, 10-Aug-2024.)
Assertion
Ref Expression
mapssfset (𝐵m 𝐴) ⊆ {𝑓𝑓:𝐴𝐵}
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem mapssfset
StepHypRef Expression
1 mapfset 8867 . . 3 (𝐵 ∈ V → {𝑓𝑓:𝐴𝐵} = (𝐵m 𝐴))
2 eqimss2 4032 . . 3 ({𝑓𝑓:𝐴𝐵} = (𝐵m 𝐴) → (𝐵m 𝐴) ⊆ {𝑓𝑓:𝐴𝐵})
31, 2syl 17 . 2 (𝐵 ∈ V → (𝐵m 𝐴) ⊆ {𝑓𝑓:𝐴𝐵})
4 reldmmap 8852 . . . 4 Rel dom ↑m
54ovprc1 7455 . . 3 𝐵 ∈ V → (𝐵m 𝐴) = ∅)
6 0ss 4392 . . 3 ∅ ⊆ {𝑓𝑓:𝐴𝐵}
75, 6eqsstrdi 4027 . 2 𝐵 ∈ V → (𝐵m 𝐴) ⊆ {𝑓𝑓:𝐴𝐵})
83, 7pm2.61i 182 1 (𝐵m 𝐴) ⊆ {𝑓𝑓:𝐴𝐵}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1533  wcel 2098  {cab 2702  Vcvv 3463  wss 3939  c0 4318  wf 6539  (class class class)co 7416  m cmap 8843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7991  df-2nd 7992  df-map 8845
This theorem is referenced by: (None)
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